In fact, analyzing the effect of dynamical friction of dark matter density spike in a binary system is not a new idea. Nevertheless, most of the related studies have focused on the binaries of the compact objects (e.g. black hole binaries) rather than the BH-LMXB systems [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}. In compact binaries, both gravitational radiation and dynamical friction of dark matter are significant. Therefore, gravitational wave detection might be required to reveal the nature of dark matter, which might contribute extra uncertainties in the constrained parameters. Since optical and X-ray observations can give very precise measurements for most of the important physical parameters in BH-LMXBs, we anticipate that analyzing BH-LMXBs can better reveal the nature of the dark matter density spike surrounding a black hole. There are at least 18 black hole X-ray binaries in our Galaxy [9]}, which can give rich information to constrain the nature of dark matter. For example, one nearby black hole X-ray binary Nova Muscae 1991 also shows an abnormally fast orbital decay \(\dot{P}=-20.7 \pm 12.7\)  ms yr\(^{-1}\) , although the uncertainty is quite large [10]}. Future high quality measurements may be helpful to further confirm the existence of dark matter density spike in these black hole X-ray binaries. This kind of analyses would open an entirely new window for observations and theoretical studies to investigate dark matter astrophysics [11]}.
<TABLE><TABLE><FIGURE>